Wingsail with adaptable flexible flap

ABSTRACT

Wingsail. The wingsail includes a substantially rigid airfoil section having a leading and a trailing edge and a flap is attached to the trailing edge through a torsion fitting having a torsional stiffness along the span of the rigid airfoil section selected to control flap deflection with respect to the rigid airfoil section under aerodynamic loading to control rolling moment of the wingsail.

This application claims priority to provisional application No.62/053,887 filed on Sep. 23, 2014, the contents of which is incorporatedherein by reference.

BACKGROUND OF THE INVENTION

This invention relates to a wingsail and more particularly to a wingsailwith a flexible flap that responds to air flow so as to reduce rollingmoment of the wingsail.

In yacht racing the geometry of the wind direction and the angle ofattack of the sails is such that aerodynamic lift on the wing isresolved into forward thrust on the yacht. The wind flow alsoestablishes a rolling moment that can cause capsizing. It isparticularly important to avoid increases in rolling moment at high windspeed to avoid dangerous capsizing.

Several researchers have considered the optimization of spanwise loadingon a wing, subject to different constraints. Jones (1) calculated theoptimum spanwise lift distribution for a wing subject to a constraint onlift and root bending moment. Tan and Wood (2) applied these ideas todetermine the optimum spanwise lift distribution for a yacht sailsubject to a constraint on the rolling moment while maximizing forwardthrust. Subsequent researchers, such as Junge et al. (3) and Sneyd andSugimoto (4) extended the analysis to include spanwise variation of windstrength and direction and boat heel. All of these analyses confirm theimportance of maximizing lift and/or forward thrust while constrainingrolling moment. In the analysis of a yacht, the geometry of the winddirection relative to the yacht direction is such that aerodynamic lifton the wing provides a component of forward thrust on the yacht. Thus wewill occasionally use the terms lift and thrust interchangeably.

These prior art analyses focused on the design of a wing or a fixed sailplanform optimized to operate at a given wind speed. As sailing hasmoved to the use of wingsails, the analysis of the sail overlaps withtraditional aerodynamics. However, unlike an aircraft wing which isdesigned to operate at a gives flight speed, and is equipped withdevices such as flaps for lower landing speeds, a racing yacht operatesover a wide range of wind speeds. Typical values would range from 10-30knots, at which point the race would be called off.

The present invention is a wingsail with a substantially rigid airfoilsection having a leading and a trailing edge. A flap is attached to thetrailing edge through a torsion fitting having a torsional stiffnessalong the span of the rigid airfoil section selected to control flapmotion with respect to the rigid airfoil section under aerodynamicloading to control rolling moment of the wingsail. In preferredembodiments, the torsional stiffness is constant along the span or thetorsional stiffness varies along the span. The flap may be segmented orunitary or both. The rigid airfoil section and the flap may have aconstant chord along the span or a varying chord along the span.

It is therefore an object of the invention to provide a wingsail havinga flap portion to control rolling moment of the wingsail.

BRIEF DESCRIPTION OF THE DRAWING

FIG. 1 is a plan view of a wing of span Y and chord due to c(y).

FIG. 2 is a cross-sectional view of a wing/flap combination.

FIG. 3 is a graph of torsional moment due to an aerodynamic moment.

FIG. 4 is a graph of total angle of attack of the flap relative to thefree stream as a function of y.

FIG. 5 is a graph of spanwise lift distribution for several windvelocities.

FIG. 6 is a graph of spanwise lift distribution at a given velocitydivided by the lift on a rigid wing having the same chord, planform andangle of attack.

FIG. 7 is a graph showing spanwise moment distribution for a constantchord wing.

FIG. 8 is a graph showing the ratio of lift and moment to that of arigid wing of the same planform.

FIG. 9 is a plan view for a general wing shape.

FIG. 10 is a graph showing spanwise variation of total angle of attackas a function of wind speed.

FIG. 11 is a graph of angle of attack for various wind speeds.

FIG. 12 is a graph of spanwise lift distribution for a variety of windspeeds at constant angle of attack.

FIG. 13 is a graph showing spanwise lift referenced to the lift of arigid wing of the same geometry.

FIG. 14 is a graph showing the contribution to rolling moment fromvarious spanwise sections.

FIG. 15 is a graph showing total lift and rolling moment relative tothat of a rigid wing as a function of wind speed.

FIG. 16 is a graph showing total lift and rolling moment relative tothat of a rigid wing for both a wing of constant chord and a wing oflinearly varying chord.

FIG. 17 is a graph of the ratio of lift/lift Ucrit against wind speedwith a constrained rolling moment.

FIG. 18 are graphs of spanwise chord and torsional stiffnessdistribution.

FIG. 19 is a graph comparing variation of angle of attack along the spanat various wind speeds in comparison with a constant chord, constanttorsional stiffness solution.

FIG. 20 shows spanwise lift distribution at a constant angle of attack.

FIG. 21 shows the span wise distribution of moment at constant angle ofattack for various wind speeds.

FIG. 22 shows results in comparison to the constant chord, constantstiffness solution.

FIG. 23 shows lift and rolling moment relative to their values from arigid wing as a function of wind speed.

FIG. 24 compares the results of total lift and rolling moment relativeto that for a rigid wing for three cases studied as a function of windspeed.

FIG. 25 is a graph showing spanwise distribution of angle of attack as afunction of non-dimensional velocity.

FIG. 26 shows spanwise distribution of lift coefficient as a function ofnon-dimensional velocity.

FIG. 27 is a graph showing spanwise distribution of rolling momentcoefficient for a range of speeds.

DESCRIPTION OF THE PREFERRED EMBODIMENT

In this patent application, we disclose the use of spanwise deformablewings to allow a wingsail to operate in an optimum and naturallyoccurring adaptive manner over a wider range of wind speeds while stillconstraining rolling moment.

Large catamarans, as are used in the America's Cup, have large wingsailswith a high aspect ratio. These wings are very effective and cangenerate significant thrust. However, these wingsails are not alwaystrimmed to give optimum performance, and due to the large span of thewind can create extreme rolling moments. In soft sails, the sail can betrimmed in order to depower the sail, and can be reefed-reducing thespan—at high wind velocities. In the wingsail case, this rolling momentcan be reduced by having multiple vertical flap sections that can beindividually controlled by the crew to give a desired span-wise twist,as described in ref (8). Altering these sections already leads to adecrease in performance, but the fact that these are manually controlledby the crew means that the sail is not trimmed at its optimizedpotential. Also, given that the crew has other responsibilities, thereare more chances for the boat to capsize in an emergency. When boatscapsize, especially boats used in the America's Cup, it is highlydangerous for the crew and can lead to substantial financial loss.

By creating a wingsail with a naturally deformable spanwise-twistingtrailing flap, we can potentially decrease this rolling moment at highwind speeds in a naturally occurring, adaptive manner without asubstantial penalty in lift and drag. Adaptive wingsails also have anadvantage in their dynamic response to sudden changes in wind speed, orgusts. The spanwise flexibility will adaptively reduce the lift on thewing during a sudden increase in wind strength. This reduction in liftwill be most pronounced at the wing tip, providing a limitation on therolling moment.

In our analysis, we consider the behavior of a wingsail consisting of aforward wing section with a flap attached to its trailing edge through atorsion fitting. The individual airfoil sections of the flap are takento be rigid in cross section but deformable in twist in the spanwisedirection. The total resistance to spanwise twist deformation inresponse to applied aerodynamic moments is characterized by thetorsional stiffness, whether this comes from a structural attachmentthat functions as a torsion rod and/or from the stiffness of the airfoilsections to spanwise twist.

We first consider a sing of constant chord, which can be solvedanalytically, then a wing of varying chord with constant torsionalstiffness; and finally a wing of varying chord with varying apply theseideas to a wingsail of any size operating in a specified range of windspeeds.

This analysis also has application to the use of wingsails to powercargo ships. Automatic reduction of rolling moment at high wind speedsdue to spanwise flexibility would be especially important for a shipthat operates on the open ocean, reducing the work load on the crewwhile maintaining good safety margins. The analysis is also applicableto extreme sailing competitions, such as round-the-world races, whichcan encounter extremely dangerous conditions off-shore.

We consider a wing of span Y and chord c(y) as shown in FIG. 1. The wingconsists of a rigid airfoil section with a flap attached to the fixedairfoil by a torsion rod of strength κ. The chord of the wing is c(y);for our calculations, we will consider a flap chord that is one-half ofthe local wing chord c_(F)(y)=½c(y). Initially, we will take thetorsional stiffness κ to be constant along the span but clearly itsvariation could be easily included, as is done in the final casestudied.

The angle of attack of the wing is taken as α; the angle of attack ofthe flap relative to the wing is α₁(y) as shown in FIG. 2. Since ourinterest is in the effect of spanwise flexibility, α₁ is a function ofy, determined by aerodynamic loads and torsional stiffness.

We will use two-dimensional linear aerodynamic theory and treat eachspanwise section using “strip” theory to estimate the effects ofspanwise flexibility on the lift and moment distribution along the span.That is, the airfoil flow is assumed to be locally two-dimensional, andthe local lift and moment are integrated along the span to obtain thetotal results for lift and rolling moment.

At each section, the aerodynamic moment about the attachment pointdepends upon both α and α₁(y).

m(y)=½ρU ² c(y)²(C _(m) ₀ α+C _(m) ₁ α₁(y))  (1)

where C_(m) ₀ and C_(m) ₁ , the local moment slope coefficients, areavailable from linear two-dimensional theory. They depend upon themagnitude of the flap chord c_(F)(y) relative to the total airfoil chordc(y). The moment on the flap acts to reduce the flap deflection α₁.

The relation between the aerodynamic moments and the local angle ofattack is given by the torsional stiffness equation.

$\begin{matrix}{{\kappa \frac{d\; \alpha_{1}}{dy}} = {{M(y)} = {\int_{y}^{Y}{{m(y)}\ {y}}}}} & (2)\end{matrix}$

where κ is the torsional stiffness: torque (in ft lbs) per radians/ft oftwist:

${K = {M/\frac{\alpha_{1}}{y}}};K$

is a local material property of the structure. M(y) is the total momentat y due to the aerodynamic moment distribution along the span. See,FIG. 3.

Since the total torsional moment M(y) goes to zero at the tip, the totaltorsional moment acting at the point y is the integral of theaerodynamic moment m(y) from y to the tip y=Y. The governing equationfor the unknown flap deflection angle α₁ is given by the derivative ofequation (2).

$\begin{matrix}{\frac{^{2}\alpha_{1}}{y^{2}} = {{m(y)} = {\frac{\rho \; U^{2}{c(y)}^{2}}{2{\kappa (y)}}\left( {{C_{m_{0}}\alpha} + {C_{m_{1}}{\alpha_{1}(y)}}} \right)}}} & (3)\end{matrix}$

This is a linear non-homogenous second order equation for α₁(y) Theboundary conditions for the equation are α₁(0)=α₁ ₀ and

$\frac{{\alpha_{1}(Y)}}{y} = 0.$

That is, the initial flap deflection α₁ ₀ , is set at the root of thewing; and since there is no moment M(y) at the tip, y=Y, the slope

$\frac{\alpha_{1}}{y}$

is zero at the tip y=Y.

The governing equation is easily solved if both the chord c(y) is aconstant: c(y)=c₀, and the torsional stiffness κ(y) is a constant,κ(y)=κ. We begin with this case. We also take the chord of the flapequal to half of the chord of the total airfoil: c_(F)=c/2. The span Yis taken as 4. For this choice, we can easily obtain the various liftand moment distributions along the wing. We will do this subsequently.

The governing equation is characterized by the ratio of the dynamicpressure times the chord c₀ ² divided by the torsional stiffness κ:written for constant chord and constant torsional stiffness we define

$Q = {\frac{\rho \; U^{2}c_{0}^{2}}{2\; \kappa}.}$

This results in the governing equation

$\begin{matrix}{{\frac{^{2}\alpha_{1}}{y^{2}} - {{QC}_{m\; 1}\alpha_{1}}} = {{QC}_{m_{0}}\alpha}} & (4)\end{matrix}$

with boundary conditions α₁(0)=α₁ ₀ and dα₁(Y)/dy=0. Defining A=C_(m) ₀Q we have

$\begin{matrix}{{\frac{^{2}\alpha_{1}}{y^{2}} - {B\; \alpha_{1}}} = {A\; \alpha}} & (5)\end{matrix}$

The solution is

$\begin{matrix}{{\alpha_{1}(y)} = \frac{\begin{matrix}{{\left( {e^{\mspace{11mu} {\sqrt{B}y}}\left( {{A\; \alpha \; t} - 1 - e^{\sqrt{B}y}} \right)} \right)*\left( {e^{\sqrt{B}y} - e^{2\sqrt{B}y}} \right)} +} \\{\alpha_{1o}B*\left( {0^{2\sqrt{B}y} + e^{2\sqrt{B}Y}} \right)}\end{matrix}}{B\left( {1 + e^{2\sqrt{B}Y}} \right)}} & (6)\end{matrix}$

From linear theory for this case, we obtain the aerodynamic momentsabout the flap attachment point due to the deflection of the forwardairfoil a and the deflection of the flap α₁(y), for an airfoil with aflap chord equal to half of the airfoil chord. These moment coefficientsare: C_(m0)α=0.212α and C_(m1)α₁(y)=0.265α₁(y). For later use, we alsoobtain the lift coefficients for the airfoil as C_(L) ₀ =2πα and C_(L) ₁=5.181α₁(y). We take the torsional stiffness κ equal to 1.

The results for the total angle of attack of the flap relative to thefree stream, α+α₁(y), as a function of y are shown in FIG. 4 for avariety of velocities U in fps. The angles α and α₁(0) are taken as 10°for all wind speeds. The effects of flap spanwise flexibility areclearly seen. At a wind speed of 60 fps, the angle of attack of the flaprelative to the free stream flow is dramatically reduced at the tip dueto its flexibility.

For a wing of constant chord c₀, the spanwise lift distribution isobtained directly from the angles of attack α and α₁(y), of the wing andthe flap, using “strip” theory. where the lift coefficients C_(L) ₀ andC_(L) ₁ have been previously introduced. The spanwise lift distributionis shown in FIG. 5 for wind velocities of 20, 30, 40, 50, and 60 fps fora torsional stiffness κ=1, α=10° and α₁ ₀ =10°. The spanwise liftdistribution for a rigid wing at 60 fps is also shown. The reduction ofspanwise lift distribution due to spanwise flexibility is dramatic.

L(y)=½ρU ² c ₀ C _(L) ₀ α+½ρU ² c ₀ C _(L) ₁ α₁(y)  (7)

FIG. 6 shows the spanwise lift distribution at a given velocity dividedby the lift on a rigid wing of the same chord, planform and angle ofattack. The effects of spanwise flexibility are clearly evident in thedecreased lift outboard of the root relative to that of a rigid wing asthe wind velocity increases.

The aerodynamic moment used to characterize the effects of spanwise flapflexibility on wing performance is the moment about the root chord, y=0:the rolling moment. For a wing of constant chord c₀, the contribution tothe rolling moment from each spanwise section y is given by

M(y)=½ρU ² c ₀(C _(L) ₀ α+C _(L) ₁ α₁(y))y  (8)

For the case examined, the results are shown in FIG. 7 for windvelocities of 20, 30, 40, 50, and 60 fps. Also shown is the contributionto the rolling moment from each spanwise section for a rigid wing. Thedecrease in sectional moment for the flexible flap is clearly evident.

The total lift and moment on the wing for a wing of constant chord c₀ isobtained by integrating the sectional lift and moment along the span.

L _(T)=½ρU ²∫₀ ^(Y) c ₀((C _(L) ₀ α+C _(L) ₁ α₁(y)dy  (9)

M _(T)=½ρU ²∫₀ ^(Y) c ₀((C _(L) ₀ α+C _(L) ₁ α₁(y)ydy  (10)

Of interest is the ratio of the total lift and total moment on theflexible wing referred to the total lift and moment on a rigid wing ofthe same planform

L _(T) _(Rigid) =½ρU ²∫₀ ^(Y) c ₀((C _(L) ₀ α+C _(L) ₁ α₁ ₀ )dy  (11)

M _(T) _(Rigid) =½ρU ²∫₀ ^(Y) c ₀((C _(L) ₀ α+c _(L) ₁ α₁ ₀ )ydy  (12)

where α₁ ₀ is the initial angle of attack of the flap at the root; for arigid wing α₁₀ remains constant along the span. The ratio of lift andmoment to that of a rigid wing of the same planform is shown in FIG. 8as a function of velocity U. The difference due to spanwise flexibilityis clearly seen, as is the more dramatic effect of flexibility on momentthan upon lift.

We now consider a wing of non-constant chord. To compare with ourearlier analytic solution, we take the root chord as c₀=1.5 and the spanas Y=4. The chord is taken with a linear variation to a tip chord ofc₁=75. The chord distribution is then c(y)=c₀−(c₀−c₁)(y/Y). The wingplanform is shown in FIG. 9.

We can also take κ to vary along the span, although for our initialcalculations we take κ=1.

The governing equation remains

$\begin{matrix}{\frac{^{2}\alpha_{1}}{y^{2}} = {{m(y)} = {{\frac{\rho \; U^{2}{c(y)}^{2}}{2\kappa \; (y)}\left( {{C_{m_{0}}\alpha} + {C_{m_{1}}{\alpha_{1}(y)}}} \right)} = {{{A(y)}\alpha} + {{B(y)}{\alpha_{1}(y)}}}}}} & (13)\end{matrix}$

with the inclusion of a chord c(y) that varies with y. The coefficientsA(y) and B(y) are as defined in equation (4) and (5), now varying withy.

${Q(y)} = \frac{\rho \; U^{2}{c(y)}^{2}}{2\kappa \; (y)}$

A(y)=C_(m) ₀ Q(y); B(y)=C_(m) ₁ Q(y).

The boundary conditions remain α₁(0)=α₁ ₀ and dy=0 at y=Y=4. For thiscase, κ is again taken as constant: κ=1.

The equation is solved numerically for α₁(y) for different values of U:U=20, 30, 40, 50, and 60 fps. The angle of attack of the wing α is takenas 10°; the initial angle of attack of the flap α₁(0) is also taken asα₁(0)=10°.

The results show the spanwise variation of the total angle α+α₁(y) as afunction of wind speed U in FIG. 10. The decrease in angle of attacktowards the tip due to spanwise flexibility at higher wing speeds isevident.

Also shown in FIG. 11 is a comparison of the constant chord case(dashed) with the varying chord numerical solution. The constant chordsolution for local angle of attack is somewhat more affected by spanwiseflexibility at lower wind speeds but the overall results are quitesimilar.

The spanwise lift and moment distribution is obtained from the striptheory formula using the solution for the flexible spanwise distortionof the flap α₁(y). For this case both α and α₁(0) were taken as 10°.

L(y)=½ρU ² c)(y)(C _(L) ₀ α+C _(L) ₁ α₁(y)  (14)

M(y)=½ρU ² c)(y)(C _(L) ₀ α+C _(L) ₁ α₁(y)  (15)

FIG. 12 shows the spanwise lift distribution L(y) for a variety of windspeeds at constant angle of attack. The effect of spanwise flexibilityat higher wind speeds is evident.

FIG. 13 shows the spanwise lift distribution referenced to the lift of arigid wing of the same geometry. Also shown dashed is the solution forthe wing of constant chord at the same root chord and span. The resultsare quite close especially at higher wind speeds.

FIG. 14 shows the contribution to the rolling moment from the variousspanwise sections. The spanwise flexibility of the flap acts to decreasethe contribution from the outboard sections.

FIG. 15 shows the total lift and rolling moment as a function of wingspeed, referred to their values for rigid wing of the same geometry. Ascan be seen, spanwise flexibility greatly reduces the lift and rollingmoment at higher wind speeds. The reduction is greater for the rollingmoment than for the lift.

Finally, the results are shown in FIG. 16 for total lift and rollingmoment for both the wing of constant chord and the wing of linearlyvarying chord, for the same value of root chord c₀=1.5 and torsionalstiffness κ=1. The results are very similar, giving the designer a toolfor designing a wing for a particular application, for example tomaintain a reasonable lift while reducing rolling moment at higher windspeeds in comparison to a rigid wing.

In the actual application of these results, the angle of attack would bereduced as the wind speed increases to maintain constraint on rollingmoment. Since the governing equations are linear, the lift and momentscale with the actual value of the angle of attack.

As an example, we assume that the constraint on rolling moment isreached at a critical wind speed of 20 fps. We then plot the ratio oflift to its value at 20 fps, using results from this case. As can beseen in FIG. 17, if the rolling moment remains constant due to changesin angle of attack, the lift continues to increase, allowing additionallift/thrust to be generated while maintaining constant rolling moment.

The previous analysis assumed constant torsional stiffness κ along thespan. Since the loading at the tip is important for the relief ofrolling moment at high wind velocities, it makes sense to examine theeffect of varying torsional stiffness κ(y) along the span. We assume alinear distribution of κ(y) as shown in the FIG. 18, with κ(0)=1; thechordwise variation of chord c(y) along the span is also shown. We allowthe torsional stiffness to decrease dramatically with spanwise distancebut do not set it to zero at the tip to avoid a singularity in thegoverning equations. The variation of the coefficients A(y) and B(y)which appear in the governing equations is also shown. Strong variationat the tip is observed.

For this case, the variation of α+α₁(y) along the span at various windspeeds is shown in FIG. 19 in comparison with the constant chord,constant torsional stiffness solution. As is expected, there is morevariation at the wing tip due to the increased tip flexibility.

Shown in FIG. 20 is the spanwise lift distribution at constant angle ofattack in comparison with the constant chord, constant torsionalstiffness solution. As expected, the lift is reduced at the wing tip athigher wind speeds.

The spanwise distribution of moment at constant angle of attack is shownin FIG. 21. The effect of spanwise flexibility acts to reduce therolling moment contribution well below that for a rood wing.

These results are shown in comparison to the constant chord, constantstiffness solution. The decrease in spanwise contribution to the totalrolling moment is evident as shown in FIG. 22.

FIG. 23 shows the lift and rolling moment relative to their values froma rigid wing as a function of wind speed. Shown is the solution forvarying chord and varying torsional stiffness, as well as the solutionfor constant chord.

FIG. 24 compares the results of total lift and rolling moment, relativeto that for a rigid wing, for the three cases studied as a function ofwind speed. These results give the designer choices to achieve a desiredoutcome.

The previous analysis was conducted for a wing of a speckle size, aswould be appropriate to predict the outcome of a wind tunnel test. It isstraightforward to extend the analysis, using non-dimensional variables,so that the results ate applicable to a wing of any size. Therequirements on the wing would specify chord c₀, span Y, operating windspeed U and desired behavior. The parameter to be identified forapplication to a specific wing is the torsional stiffness κ: torque Mrequired in ft lbs per to produce a twist dα/dy, in radians/ft.

We consider the case of constant chord c₀ for analytic simplicity; themore general case can easily be considered. We begin our analysis withequation (3).

$\begin{matrix}{\frac{^{2}\alpha_{1}}{y^{2}} = {{m(y)} = {\frac{\rho \; U^{2}{c(y)}^{2}}{2\kappa \; (y)}\left( {{C_{m_{0}}\alpha} + {C_{m_{1}}{\alpha_{1}(y)}}} \right)}}} & (16)\end{matrix}$

This equation is written for a wing span from y=0 to y=Y where Y is thewing span. We nondimensionalize the equation using the variable y′=y/Y.The equation becomes

$\begin{matrix}{\frac{^{2}\alpha_{1}}{y^{2}} = {{m(y)} = {\frac{\rho \; U^{2}c_{0}^{2}Y^{2}}{{2\kappa}\;}\left( {{C_{m_{0}}\alpha} + {C_{m_{1}}{\alpha_{1}\left( y^{\prime} \right)}}} \right)}}} & (17)\end{matrix}$

This allows us to identity the governing non-dimensional parameter whichwe call Ū.

$\begin{matrix}{\overset{\_}{U} = \sqrt{\frac{\rho \; U^{2}c_{0}^{2}Y^{2}}{{2\kappa}\;}}} & (18)\end{matrix}$

We rewrite the equation as

$\begin{matrix}{\frac{^{2}\alpha_{1}}{y^{2}} = {{m(y)} = {{\overset{\_}{U}}^{2}\left( {{C_{mo}\alpha} + {C_{m\; 1}{\alpha_{1}\left( y^{\prime} \right)}}} \right)}}} & (19)\end{matrix}$

The equation now contains one non-dimensional variable Ū and the twomoment coefficients that have already been introduced. The solutionfollows as before.

We construct the solution for the flap angle α₁(y′) for constant chord,since this results in a simple analytic solution.

$\begin{matrix}{{\alpha_{1}\left( y^{\prime} \right)} = \frac{\begin{matrix}{e^{{- \sqrt{C_{m1}}}{Uy}^{\prime}}\left( {{{- \alpha}\; {C_{m\; 0}\left( {e^{2\sqrt{C_{m\; 1}}U} - e^{\sqrt{C_{m\; 1}}{Uy}^{\prime}}} \right)}\left( {{- 1} + e^{\sqrt{C_{m\; 1}}{Uy}^{\prime}}} \right)} +} \right.} \\\left. {\alpha_{1\; o}{C_{m\; 1}\left( {e^{2\sqrt{C_{m\; 1}}U} + e^{2\sqrt{C_{m\; 1}}{Uy}^{\prime}}} \right)}} \right)\end{matrix}}{C_{m\; 1}\left( {1 + e^{2\sqrt{C_{m\; 1}}U}} \right)}} & (20)\end{matrix}$

This equation determines α₁(y′) for various values of thenon-dimensional parameter Ū. The results for the total angle of attackα+α₁(y) are shown in FIG. 25. The reduction in angle of attack due tospanwise flexibility is quite pronounced in the range 8<Ū.

The spanwise distribution of lift coefficient C_(L)(y′) is shown in FIG.26 for a range of Ū from 4 to 16. The range 8<Ū shows a dramaticdecrease in the contribution of outboard wing sections to the rollingmoment coefficient due to spanwise flexibility.

The spanwise distribution of rolling moment coefficient C_(M)(y′) isshown in FIG. 27 for a range of Ū from 4 to 16. The range 8<Ū shows adramatic decrease in the contribution of outboard wing sections to therolling moment coefficient due to spanwise flexibility.

We now consider how these non-dimensional results relate to our earliercalculations for a specific planform. We consider the constant chordsolution: Y=4; c₀=1.5; and k=1 and U=40 fps. This results in a Ū=8.27.(√{square root over (ρ(40 fps)²*4²1.5²/(2*(κ=1))}{square root over (ρ(40fps)²*4²1.5²/(2*(κ=1))}=Ū=8.27.)

We then inquire as so what value of κ would be required to realize thissame result (distribution of α+α₁(y)) for a wing of span 70 ft., with achord of 10 ft. at a wind speed of 25 kts. For this case, we set√{square root over (ρ(25 kts/0.59)²*70²*10²/(2*κ))}{square root over(ρ(25 kts/0.59)²*70²*10²/(2*κ))}=Ū=8.27 and obtain κ=15288. (The factor0.59 is the conversion from kts to fps.)

Once the spanwise deflection of the flap α₁(y) is determined for a givenwind speed, the aerodynamic properties of lift and moment along the spanas well as the total lift and total rolling moment can be determined.

As previously noted, at higher wind speeds the angle of attack can bereduced to constrain rolling moment while the lift continues toincrease, increasing thrust,

REFERENCES

-   (1) Jones, R. T. “The Spanwise Distribution of Lift for Minimum    Induced Drag of Wings Having a Given Lift and a Given Bending    Moment” NACA TN2249, 1950-   (2) Wood, C. J., Tan, S. H., “Towards an optimum yacht sail.”    Journal of Fluid Dynamics, Vol. 85, Part 3, 1978, pp. 459-477.-   (3) Junge, Timm, Gerhardt, Frederik C., Richards, Peter, Flay,    Richard G. J., “Optimization Spanwise Lift Distributions Yacht Sails    Using Extended Lifting Line Analysis,” Journal of Aircraft, Vol. 47,    No. 6, November-December 2010.-   (4) Sneyd, A. D., Sugimoto, T., “The influence of a yacht's heeling    stability on optimum sail design,” Fluid Dynamics Research, Vol.    19, 1997. pp. 47-63.-   (5) Harmon, Robyn Lynn, Aerodynamic Modeling of a Flapping Membrane    Wing Using Motion Tracking Experiments, ProQuest LLC, Ann Arbor,    Mich., 2009-   (6) Fisher, Adam, “The Boat That Could Sink the America's Cup,”    Wired, May 9, 2013. http://www.wired.com/2 cup-boat-crash/. Accessed    Mar. 31, 2014.-   (7) Fisher, Adam, “What Went Wrong in the Deadly America's Cup    Crash,” Wired, May 9, 2013,    http://www.wired.com/2013/05/americas-cup-crash/. Accessed Apr. 3,    2014.-   (8) http://www.cupexperience.com/americas-cup-ac72-design-wing-sail/

What is claimed is:
 1. Wingsail comprising: a substantially rigidairfoil section having a leading and a trailing edge; and a flapattached to the trailing edge through a torsion fitting having atorsional stiffness along the span of the rigid airfoil section selectedto control flap motion with respect to the rigid airfoil section underaerodynamic loading to control rolling moment of the wingsail.
 2. Thewingsail of claim 1 wherein the torsional stiffness is constant alongthe span.
 3. The wingsail of claim 1 wherein the torsional stiffnessvaries along the span.
 4. The wingsail of claim 1 wherein the flap issegmented.
 5. The wingsail of claim 1 wherein the rigid airfoil sectionhas a constant chord along the span.
 6. The wingsail of claim 1 whereinthe rigid airfoil section has a varying chord along the span.